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Decomposition of \( S_{8}^{\mathrm{new}}(9) \) into irreducible Hecke orbits

magma: S := CuspForms(9,8);
magma: N := Newforms(S);
sage: N = Newforms(9,8,names="a")
Label Dimension Field $q$-expansion of eigenform
9.8.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(6q^{2} \) \(\mathstrut-\) \(92q^{4} \) \(\mathstrut-\) \(390q^{5} \) \(\mathstrut-\) \(64q^{7} \) \(\mathstrut+\) \(1320q^{8} \) \(\mathstrut+O(q^{10}) \)
9.8.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(232q^{4} \) \(\mathstrut-\) \(16 \alpha_{2} q^{5} \) \(\mathstrut+\) \(260q^{7} \) \(\mathstrut+\) \(104 \alpha_{2} q^{8} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{10}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 360\)

Decomposition of \( S_{8}^{\mathrm{old}}(9) \) into lower level spaces

\( S_{8}^{\mathrm{old}}(9) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/3/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(3)) }^{\oplus 2 } $